text:
\section{Introduction}

The study of exoplanets has evolved from detection to characterization, thanks to the advent of cutting-edge observational techniques.  The spectra of exo-atmospheres provide us with valuable clues about the atmospheric chemistry and thermal structure.  Diagnosing and interpreting these spectra to obtain chemical compositions is now at the forefront of exo-atmospheric research.

The simplest assumption is to build a model in chemical equilibrium, where the molecular composition for a given elemental abundance only depends on local, basic parameters (pressure and temperature) independent of the reaction pathways. However, equilibrium chemistry only holds in the hot ($T \gtrsim 2000$ K) or deep ($P \gtrsim 100$ bar) parts of the atmosphere.  Processes like ultraviolet (UV) irradiation and atmospheric dynamics drive the chemical composition in the observable atmosphere away from equilibrium. These disequilibrium processes commonly dominate the observable parts of atmospheres in the Solar System.

Chemical kinetics models, which incorporate a chemical network of hundreds to thousands of reactions, are needed to study disequilibrium chemistry (e.g., \citealt{koppa12,moses11,venot12,krome14,venot12,hu14,rm16,tsai17}).  Even with such a number of reactions, chemical networks are greatly reduced compared to Nature and restricted to measurements in specific temperature ranges.  In other words, there does not exist a one-fits-all chemical network in practice.  The accuracy of a network is determined by whether it includes the relevant reactions and if the input rate coefficients are reliable. Unfortunately, these chemical kinetics models are expensive to run, especially when one desires to couple chemistry to three-dimensional atmospheric motion.  In one dimension, eddy diffusion is used to mimic large-scale atmospheric circulation, convection, turbulence, etc.  The three-dimensional atmospheric circulation patterns of tidally-locked, highly-irradiated exoplanets are demonstrably more complicated (e.g., \citealt{showman09,dobbs13} and see \citealt{hs15} for a review).  In order to correctly interpret the transmission spectra of hot Jupiters, there is a need to develop three-dimensional general circulation models (GCMs) that couple the atmospheric dynamics and chemistry.  Intermediate steps have already been taken in this direction by, e.g., \cite{agu12,agu14}, who coupled a chemical kinetics code to a simplified model of the atmospheric dynamics (constant solid-body rotation mimicking a uniform equatorial jet). There is clearly a need to build on studies like these, but a brute-force coupling between a three-dimensional solver of the fluid equations and a chemical kinetics code with a network of hundreds to thousands of reactions is computationally challenging, even without considering radiative transfer (which is needed to include photochemistry).

Another approach is to simplify the chemical scheme. Conceptually, the interaction between atmospheric motion and chemistry is a comparison between two timescales: the dynamical versus chemical timescales.  The simplest approach is to use the timescale argument, or quenching approximation.  It assumes that the deep atmosphere is in chemical equilibrium, because the chemical timescale is much shorter than the dynamical timescale.  There is a location within the atmosphere known as the quench point, where the timescales are equal.  Above the quench point, the chemical abundances are assumed to be well-mixed and frozen to their equilibrium values at the quench point. In technical parlance, the process is referred as ``transport-induced quenching". The quenching approximation has been used to understand the over-abundance of carbon monoxide (CO) in the upper troposphere of Jupiter \citep{prinn}.  \cite{vc12} compared the chemical timescale for converting methane (CH$_4$) to CO to the orbital timescales of highly eccentric exoplanets to study the interaction between the evolving thermal structures and the chemistry.  While the quenching approximation can be used to build the bulk of our intuition, it has been shown that it should be applied with caution to know when the abundance of a molecule is controlled by the disequilibrium abundance of another parent molecule, e.g., acetylene being controlled by methane \citep{tsai17}. Additionally, the quenching approximation contains the ambiguity of having to specify an appropriate length scale, which is not known from first principles \citep{smith98}.

A more realistic approach is chemical relaxation. It was pioneered in the exo-atmospheres literature by \cite{cs06}.  Instead of simplifying the treatment of atmospheric dynamics, chemical relaxation takes the approach of replacing the chemical network with a single source/sink term that depends on the chemical timescale.  \cite{cs06} coupled chemical relaxation to a simplified GCM to study quenching of CO, and suggested that most of the carbon is locked up in CO in HD 209458b due to transport. However, a shortcoming of their study is the assumption of a single rate-limiting reaction for the interconversion and probably the underestimation of the timescale of CO.

A prerequisite for implementing chemical relaxation is to be able to compute the chemical timescale over a broad range of temperatures and pressures.  Initially, we had hoped to seek a universal fit for the chemical timescale across temperature, pressure and metallicity, motivated by the work of \cite{zm14}.  Figure \ref{fig:fit_ZM} shows our attempt at fitting an Arrhenius-like function to the chemical timescale for CH$_4$-CO conversion. We find the timescales explored in \cite{zm14} are likely restricted to a narrow temperature-pressure range. It is apparent that such a simple approach fails to fit the timescale across the range of values of temperature and pressure needed for us to implement chemical relaxation.

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width=\columnwidth]{f1.eps}
\end{center}
\vspace{-0.2in}
\caption{Chemical timescales associated with methane for 1500 K (blue solid curve) and 2000 K (blue dashed curve) computed using full chemical kinetics.  Following Figure 5 of \cite{zm14} $^1$, we found a fit for $\tau_{\rm CH_4}$ in the range of $1300 \mbox{ K} \lesssim T \lesssim 2200$ K and $0 \lesssim P \lesssim 100$ bar: $5.6 \times 10^{-11} p^{-0.53} e^{52759 \mbox{ (K)}/T}$.  We then overlaid this Arrhenius-like fit (black, solid and dashed curves) with full chemical-kinetics calculations of $\tau_{\rm CH_4}$, over a broader range of pressures, in  for two values of the temperature (1500 and 2000 K).}
\label{fig:fit_ZM}
\end{figure}
\footnotetext[1]{Note that $\tau_{\ce{CO}}$ is not distinguished from $\tau_{\ce{CH4}}$ in \cite{zm14}: these two timescales are only equal when [CO]/[\ce{CH4}] = 1. We find in fact the left panel in Figure 5 of \cite{zm14} shows $\tau_{\ce{CH4}}$ and the right panel shows $\tau_{\ce{CO}}$ due to different temperatures.}

Generally, the chemical timescale is a \textit{function} that draws upon different rate-limiting reactions in different temperature and pressure regimes. A challenge with the chemical relaxation method is to find these rate-limiting reactions. In doing so, we develop a simple method to identify the dominant pathway and the associated rate-limiting step (RLS) in the chemical network for given values of the temperature, pressure, and elemental abundances. The pathway analysis can be used for comparing chemical kinetics calculations performed by different groups using different chemical networks, as it allows one to identify the key reactions that essentially control the output, and assess if key reactions are missing. It is useful both on from a practical point of view and for developing physical intuition (e.g., \citealt{thomas}).

Ever since the work of \cite{cs06}, only \cite{ben18} has recently implemented the relaxation method in the Met Office Unified Model and improved it by consistently coupling to radiative transfer. However, to our knowledge, chemical relaxation has surprisingly never been validated. By ``validation", we mean that the accuracy of chemical relaxation should be demonstrated to a factor of a few, rather than to $\sim 1\%$ accuracy (or better), given the existing uncertainties in the rate coefficients and the approximate nature of the approach. In addition, transit radii are proportional to the logarithms of the chemical abundances, such a factor-of-several validation suffices for studying atmospheric chemistry in hot Jupiters.  In the current study, we perform this validation step in one dimension, since it is a test of the ability of the chemical relaxation scheme to mimic the full chemical network, rather than one of the complicated three-dimensional geometry.  In a future work, we will aim to couple the chemical relaxation scheme to a three-dimensional GCM, but in the current work we restrict ourselves to validating the scheme on a factor-of-several basis.  Such a strategy follows the well-established hierarchical approach of constructing climate models \citep{held05}.  We do not consider photochemistry for the current study.

In \S\ref{sec:theory}, we provide the background theory on how to compute the chemical timescale.  In \S\ref{sec:rls}, we describe our methodology for identifying the rate-limiting chemical reactions.  In \S\ref{sec:chemtime}, we describe how we compute the chemical timescales.  In \S\ref{sec:pathway}, we compare our chemical network to that of Moses and Venot using our pathway analysis tool.  In \S\ref{sec:validate}, we validate the chemical relaxation method for three model atmospheres.  In \S\ref{sec:summary}, we summarize our findings and list opportunities for future work.

\section{Simplified expression for reaction rate equations}
\label{sec:theory}

To compute the chemical timescale for use in the chemical relaxation method, we need to first derive a simplified expression for it that depends only on the local conditions of temperature and  pressure.  This allows us to evaluate the rate of change of the abundances efficiently.

Consider a chemical species subject to production (${\cal P}$)  and loss (${\cal L}$) through a network of chemical reactions without any disequilibrium process (e.g., transport).  The rate of change of its volume number density ($n$) is
\begin{equation}
\frac{d n}{d t} = {\cal P} - n {\cal L}.
\label{eq:master}
\end{equation}
Equation (\ref{eq:master}) is written in a way that ${\cal P}$ and ${\cal L}$ do not depend on $n$, as production only depends on other species and loss depends linearly on the number density of the reactant for a typical bimolecular reaction. Since ${\cal P}$ and ${\cal L}$ depend on the number densities of other species, the ensemble of equation (\ref{eq:master}) for every species forms a system of coupled differential equations and the calculation involves inverting a large matrix (e.g., \citealt{hu12,tsai17}). If one wishes to implement chemical kinetics into the dynamical core of a GCM, then one needs to include a separate Euler equation for every species in the chemical network.

The relaxation method rewrites equation (\ref{eq:master}) as \citep{smith98,cs06}
\begin{equation}
\frac{dn}{dt} = -\frac{n-n_\mathrm{EQ}}{\tau_\mathrm{chem}},
\label{eq:relax}
\end{equation}
replacing the production and loss terms with a source/sink term that relaxes $n$ to $n_\mathrm{EQ}$ by the chemical timescale ($\tau_{\rm chem}$). When the abundance is greater (less) than its equilibrium value, then it decreases (increases) and works its way towards equilibrium. The chemical timescale is effectively determined by the employed chemical network. When coupled to a GCM, whether the species in question attains chemical equilibrium depends on the competition between atmospheric dynamics and chemistry via their corresponding timescales.  Chemical relaxation is analogous to the Newtonian relaxation method used as a substitute for radiative transfer (e.g., \citealt{hs94}) or the treatment of condensation where the supersaturated gas is relaxed to the saturated vapor number density on the condensation timescale \citep{hu12}.

The chemical timescale is conventionally expressed without justification as
\begin{equation}
\tau_\mathrm{chem} = \frac{n}{\vert dn/dt \vert} = \frac{n_\mathrm{EQ}}{n^\prime_\mathrm{EQ} {\cal L}^\prime_\mathrm{EQ}},
\label{eq:tau}
\end{equation}
where the second equality assumes that the species in question has a number density that is equal to its chemical equilibrium value.  Furthermore, $n^\prime_\mathrm{EQ} {\cal L}^\prime_\mathrm{EQ}$ refers to the loss rate determined by the RLS involving other species but not $n$, denoted with primes (The ambiguity comes from using the equilibrium abundance for the numerator but not the denominator in equation(\ref{eq:tau}) since $dn/dt$ vanishes in chemical equilibrium).

We now wish to demonstrate that equation (\ref{eq:tau}) may be derived from equation (\ref{eq:master}) and (\ref{eq:relax}).  First, consider the situation when $n \ll n_{\rm EQ}$.  This implies that the loss of the species being considered is negligible compared to production, which means $dn/dt \approx {\cal P}$.  If we assume that all of the other species in the network are close to chemical equilibrium (as most of the intermediate species are fast-reacting radicals), then we can further write ${\cal P} \approx {\cal P}^\prime_{\rm EQ}$ because recall that ${\cal P}$ does not depend on $n$.  For the RLS, we can write ${\cal P}^\prime_{\rm EQ} = n^\prime_{\rm EQ} {\cal L}^\prime_{\rm EQ}$.  By inserting this expression into equation (\ref{eq:master}) and (\ref{eq:relax}), we obtain equation (\ref{eq:tau}).  In the opposite limit of $n \gg n_{\rm EQ}$, the loss of the species dominates production and we have $dn/dt = -n {\cal L} \approx - n^\prime_{\rm EQ} {\cal L}^\prime_{\rm EQ}$.  This again leads to equation (\ref{eq:tau}) from equation (\ref{eq:master}) and (\ref{eq:relax}). Since the chemical timescale expression is approximately correct for both limits, it is expected to work at order-of-magnitude accuracy  at least. This expectation will be confirmed by full numerical calculations of chemical kinetics.

\section{Determining the rate-limiting reactions in the chemical network}
\label{sec:rls}
It is common for the chemical conversion of one species to another to not occur in one step. Rather, it takes multiple steps to surmount the energy barrier via the breaking or forming of chemical bonds.  These sequence of reactions form a pathway, and the chemical timescales associated with each step in the pathway may differ by many orders of magnitude.  The efficiency of a pathway is bottlenecked by its slowest reaction.  Specifically, \textit{the RLS is defined as the slowest reaction along the fastest pathway.}  It informs the effective loss rate, $n^\prime_\mathrm{EQ} {\cal L}^\prime_\mathrm{EQ}$, in equation (\ref{eq:tau}).  Thus, computing the chemical timescale involves identifying the RLS.  For example, \cite{zm14} have remarked how the conversion of CO to CH$_4$ may be visualized as the reduction of the bond between C and O from a triple bond to a double bond to a single bond and eventually splitting C from O, in three steps: first between CO and formaldehyde (H$_2$CO), second between formaldehyde and methanol (CH$_3$OH) and finally between methanol and methane. We build upon and extend the diagram in Figure \ref{fig:scheme}, where the temperature-and-pressure dependent pathways and RLSs are included, as explained in the following subsections.

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width=\columnwidth]{f2.pdf}
\end{center}
\vspace{-0.2in}
\caption{Visualization of the major chemical pathways between methane and carbon monoxide in hydrogen-dominated atmospheres.  The triple, double and single bonds between carbon and oxygen are colored red, orange and green, respectively.  The blue arrows represent the pathway at high temperatures and low pressures for $\mbox{C/O} < 1$.  The brown arrows represent the pathway turned up for $\mbox{C/O} > 1$.  For a description of the (\ref{R1}) to (\ref{R10}) reactions, see text. For their specific operating temperatures and pressures, see Tables \ref{tab: pathways} (solar abundance) and \ref{tab: pathways_CtoO2} (C/O=2).}
\label{fig:scheme}
\end{figure}

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width=\columnwidth]{f3.eps}
\end{center}
\vspace{-0.2in}
\caption{Parameter space of temperature and pressure showing how the rate-limiting step corresponds to different chemical reactions (R1 to R9; for more details, see text).  These 9 chemical reactions may in turn be visualized as belonging to 3 different schemes (A, B and C; see text).}
\label{fig: rls}
\end{figure}

\subsection{CH$_4$-CO interconversion}

\subsubsection{Identifying the rate-limiting steps from full chemical kinetics}

In addition to the RLS being a function of temperature and pressure, extracting this information to identify it is not straightforward because the possible number of paths grow exponentially with increasing number of species, making it difficult to track them all. Our approach is to develop a tool using Dijkstra's algorithm to find the shortest path and the associated RLS; see Appendix \ref{app:pathway} for more details.

Figure \ref{fig: rls} shows a survey of the different RLSs as functions of temperature and pressure and for protosolar elemental abundances from \citet{lodders09} ($\mbox{C/H}=2.776 \times 10^{-4}, \mbox{O/H}=6.062 \times 10^{-4}, \mbox{N/H}=8.185 \times 10^{-5}, \mbox{He/H} = 9.69 \times 10^{-2}$).  Unsurprisingly, there does not exist a single RLS for the range of temperatures (500--3000 K) and pressures (0.1 mbar--1 kbar) considered. The different reactions, labeled R1 to R10, are
\begin{reactions}\label{R1}
\ce{OH + CH3 + M -> CH3OH + M}
\end{reactions}
\begin{reactions}\label{R2}
\ce{OH + CH3 -> CH2OH + H}
\end{reactions}
\begin{reactions}\label{R3}
\ce{CH2OH + M -> H + H2CO + M}
\end{reactions}
\begin{reactions}\label{R4}
\ce{CH3 + O -> H2CO + H}
\end{reactions}
\begin{reactions}\label{R5}
\ce{OH + C -> CO + H}
\end{reactions}
\begin{reactions}\label{R6}
\ce{H + CH4 -> CH3 + H2}
\end{reactions}
\begin{reactions}\label{R7}
\ce{CH3OH + H -> CH3O + H2}
\end{reactions}
\begin{reactions}\label{R8}
\ce{C2H2 + O -> CH2 + CO}
\end{reactions}
\begin{reactions}\label{R9}
\ce{CH3 + H2O -> CH3OH + H}
\end{reactions}
\begin{reactions}\label{R10}
\ce{C2H2 + OH -> CH3 + CO}
\end{reactions}
where M refers to any third body (i.e. the total number density of the gas). Reaction R8 and R10 are only relevant when $\mbox{C/O}>1$, as we will describe shortly in the example of $\mbox{C/O}=2$.

\subsubsection{Grouping of reactions into three schemes}

It is possible to understand CH$_4$-CO interconversion as consisting of three schemes (at least, for solar-like elemental abundances). As temperature increases, the scheme moves from (C) to (A), as higher kinetic energy allows more ambitious steps.:
\begin{itemize}

\item (A): Though the progressive dehydrogenation of CH$_4$ into C, followed by oxidization into CO (blue arrows in Fig \ref{fig:scheme});

\item (B): Via H$_2$CO (formaldehyde) from directly oxidizing CH$_3$ as described by (\ref{R4});

\item (C): Via intermediate species like CH$_2$OH, CH$_3$OH, or CH$_3$O from oxidizing CH$_3$ (through the molecules shown in green in Fig \ref{fig:scheme}).

\end{itemize}
At high temperatures and low pressures (the magenta region in Figure \ref{fig: rls}), scheme (A) is favored because it requires a high abundance of atomic hydrogen, produced mainly by thermal decomposition of H$_2$. Scheme (B) sits in the transition between scheme (A) and (C).  Scheme (C) covers the broadest range of temperature and pressure and contains the pathways previously identified by, e.g., \cite{yung88,bezard,moses11,vc12}. As shown in Figure \ref{fig:scheme}, CH$_4$ is first converted to CH$_3$ before being oxidized by OH or H$_2$O to form CH$_2$OH or CH$_3$OH/CH$_3$O depending on the temperature and pressure.  These singly-bonded (\ce{C-O}) intermediate species make forming the double bond (\ce{C=O}) in H$_2$CO easier than directly from C and O.  H$_2$CO goes on to efficiently produce HCO and finally the triple-bonded structure of CO.  At high enough temperatures ($T \gtrsim 2000$ K), there is sufficient energy to directly form the double bond between C and O into H$_2$CO via reaction (\ref{R4}) without passing through the intermediate species, which is scheme (B).  This is similar to the RLS, \ce{CH3 + OH -> H2 + H2CO}, initially suggested by \cite{prinn} to explain the quenched CO found in Jupiter.  At even higher temperatures, if the pressure remains low enough then scheme (A) dominates.  Molecular hydrogen is dissociated into atomic hydrogen, which in turn promotes the dehydrogenation of methane.  Eventually, the accumulated C is present at high enough abundances that allow for its oxidization into CO.

A typical pathway of scheme (A) is:
\begin{subequations}
\label{re:A}
\begin{align}
\begin{split}
\ce{ CH4 + H} &\rightarrow \ce{ CH3 + H2 } \\
\ce{ CH3 + H} &\rightarrow \ce{ CH2 + H2  } \\
\ce{ CH2 + H} &\rightarrow \ce{ CH + H2  } \\
\ce{ CH + H} &\rightarrow \ce{ C + H2 } \\
\ce{ C + OH} &\rightarrow \ce{ CO + H  } \\
\ce{ H + H2O}&\rightarrow \ce{ OH + H2 } \\
\ce{ 2( H2 + M}&\rightarrow \ce{ 2H + M )} \\
\hline \nonumber
\mbox{net} : \ce{ CH4 + H2O} &\rightarrow \ce{CO + 3H2 }
\end{split}
\end{align}
\end{subequations}

Scheme (B), which does not involve intermediate species like \ce{CH3OH} or \ce{CH2OH}, goes through the pathway:
\begin{subequations}
\label{re:B}
\begin{align}
\begin{split}
\ce{ CH4 + H} &\rightarrow \ce{ CH3 + H2 } \\
\ce{ CH3 + O} &\rightarrow \ce{ H2CO + H  } \\
\ce{ H2CO + H} &\rightarrow \ce{ HCO + H2 } \\
\ce{ HCO + H} &\rightarrow \ce{ CO + H2 } \\
\ce{ OH + H} &\rightarrow \ce{ O + H2 } \\
\ce{ H + H2O }&\rightarrow \ce{ OH + H2 } \\
\ce{ 2( H2 + M}&\rightarrow \ce{ 2H + M )} \\
\hline \nonumber
\mbox{net} : \ce{ CH4 + H2O} &\rightarrow \ce{CO + 3H2 }
\end{split}
\end{align}
\end{subequations}

Scheme (C) includes several main pathways for $T$ $\lesssim$ 2000 K or P $\gtrsim$ 1 bar. One example of scheme (C) is:
\begin{subequations}
\label{re:C}
\begin{align}
\begin{split}
\ce{ CH4 + H} &\rightarrow \ce{ CH3 + H2 } \\
\ce{ CH3 + OH + M}&\rightarrow \ce{ CH3OH + M  } \\
\ce{ CH3OH + H }&\rightarrow \ce{ CH3O + H2 } \\
\ce{ CH3O + M }&\rightarrow \ce{ H2CO + H + M } \\
\ce{ H2CO + H }&\rightarrow \ce{ HCO + H2 } \\
\ce{ HCO + M }&\rightarrow \ce{ H + CO + M } \\
\ce{ H + H2O }&\rightarrow \ce{ OH + H2 } \\
\ce{ H2 + M }&\rightarrow \ce{ 2H + M }\\
\hline \nonumber
\mbox{net} : \ce{ CH4 + H2O} &\rightarrow \ce{CO + 3H2 }
\end{split}
\end{align}
\end{subequations}
More examples of pathways belonging to scheme (C) may be found in \cite{moses11,rm16,tsai17}.

\subsubsection{Comparison with previous work}

Previous work studying CO-CH$_4$ interconversion has typically assumed one or two RLS, because their intentions were to estimate the location of the quench point \citep{yung88,lf02,ms11,vm11}.  For the purpose of implementing chemical relaxation, this is insufficient.  In Figure \ref{fig:tau_contours}, we demonstrate this by comparing the calculations of chemical timescales associated with CH$_4$ and CO with those from \cite{cs06} and \cite{vc12}. The discrepancies between our calculations and those from \cite{vc12} are mainly present at high temperatures and low pressures. There are significant discrepancies between our calculations and the approximate ones by \cite{cs06} due to a different RLS assumed in \cite{cs06} as explained in the following paragraph. We use equation (16), instead of (19), of \cite{cs06}, because the latter is an approximation that is valid only when $T \lesssim 2000$ K and the mole fraction of H$_2$ is close to 1.  Instead of using equations (1) to (5) from \cite{cs06}, we perform the full chemical-equilibrium calculations.  We also include chemical timescales associated with NH$_3$ and N$_2$ for completeness.

\begin{figure*
\begin{center}
\vspace{-0.1in}
\includegraphics[width=\columnwidth]{f4a.eps}
\includegraphics[width=\columnwidth]{f4b.eps}
\includegraphics[width=\columnwidth]{f4c.eps}
\includegraphics[width=\columnwidth]{f4d.eps}
\end{center}
\vspace{-0.2in}
\caption{Chemical timescales associated with the production of CH$_4$, CO, NH$_3$ and N$_2$ as functions of temperature and pressure.  The black curves are our calculations, while the cyan curves are from \cite{vc12} (for CH$_4$ and CO) and the yellow curves are from \cite{cs06} (for CO).}
\label{fig:tau_contours}
\end{figure*}

\cite{cs06} consider a single CO-\ce{CH4} pathway
\begin{subequations}
\begin{align}
\ce{H2CO + H + M} &\rightarrow \ce{CH3O + M} \label{Ra}\\
\ce{CH3O + H2} &\rightarrow \ce{CH3OH + H} \label{Rb}\\
\ce{CH3OH + H} &\rightarrow \ce{CH3 + H2O} \label{Rc}
\end{align}
\end{subequations}
proposed by \cite{yung88}, where CO reacts with hydrogen to form \ce{H2CO} (formaldehyde)  and goes through \ce{CH3O} (methoxide) and \ce{CH3OH} (methanol) to get to \ce{CH3} (methyl).
They suggested (\ref{Ra}), which is involved in breaking the \ce{C=O} bond, as being the RLS.  \cite{cs06} adopt the rate constants from \cite{page} and \cite{bezard} for the low- and high-pressure limits, respectively.  We verify that the rate coefficient from \cite{page} has a value that is similar to what we use in our chemical network and is not the source of the discrepancies between our calculations and those of \cite{cs06}.

Rather, the discrepancies stem from the rate coefficient associated with (\ref{Rb}), which had not been measured experimentally. \cite{yung88} assumed this reaction to be relatively fast, based on comparison with other similar reactions (see their Appendix A).  Motivated by the importance of CH$_3$OH kinetics (see \citealt{vc10} for details), \cite{moses11} performed ab initio calculations for the rate coefficients. According to their rate coefficients, (\ref{Rb}) always reacts slower than (\ref{Rc}) and thus should be the RLS \footnotemark[2] We have chosen to use the rate coefficients of \cite{moses11}. \footnotetext[2]{We also find this pathway suggested by \cite{yung88}, except for (\ref{Rb}) being the RLS instead of (\ref{Ra}), becomes dominant at high pressures ($\gtrsim$ 100 bar) and is important for Jupiter and Saturn where the \ce{CH4}-CO quench level is much deeper.}

\cite{vc12} identified (\ref{R1}) and (\ref{R2}) as being the RLSs and adopt the rate coefficients from \cite{jasper}.  We have also taken the rate coefficients for (\ref{R1}) from \cite{jasper}, but the reverse rate coefficient for (\ref{R2}) from \cite{tsang}.  The differences between these rate coefficients are within a factor of 2. The two RLSs in \cite{vc12} control the most relevant temperature-pressure regions for hot/warm Jupiters ($\sim$ 1000 - 1500 K) and their timescale agrees well with our calculation until entering the high-temperature and low-pressure regime.

\subsection{NH$_3$-N$_2$ interconversion}

\begin{figure}[h]
\begin{center}
\vspace{-0.1in}
\includegraphics[width=0.9\columnwidth]{f5.pdf}
\end{center}
\vspace{-0.2in}
\caption{Schematic illustration as Figure \ref{fig:scheme} but for the major chemical pathways between \ce{NH3}, \ce{N2}, and HCN.}
\label{fig:N_scheme}
\end{figure}

The timescales of nitrogen species are less constrained than that of CH$_4$-CO interconversion. HCN participates when $T$ $\gtrsim$ 1000 K and complicates the interconversion. We find it not straightforward to quantify the contribution of HCN and the real timescales deviate from those simply considering NH$_3$-N$_2$ interconversion.

The RLSs for NH$_3$-N$_2$ interconversion are
\begin{reactions}\label{R11}
\ce{NH2 + NH2 -> N2H2 + H2}
\end{reactions}

\begin{reactions}\label{R12}
\ce{N2H3 + M -> N2H2 + H + M}
\end{reactions}

\begin{reactions}\label{R13}
\ce{NH + NH2 -> N2H2 + H}
\end{reactions}

\begin{reactions}\label{R14}
\ce{NO + NH2 -> N2 + H2O}
\end{reactions}

\begin{reactions}\label{R15}
\ce{N + NO -> N2 + O}
\end{reactions}

\begin{reactions}\label{R16}
\ce{N2H4 + H -> N2H3 + H2}
\end{reactions}

Figure \ref{fig:N_scheme} visualizes the network of RLSs for nitrogen chemistry, while Table \ref{tab: pathways_N2} lists the RLSs of NH$_3$-N$_2$, across temperature and pressure.

NH$_3$-N$_2$ interconversion can be divided into two schemes, depending on whether \ce{N2} is formed from \ce{N2H} or NO. At high pressures (the parameter space occupied by (\ref{R11}), (\ref{R12}), and (\ref{R13}) in Table \ref{tab: pathways_N2}), \ce{N2} is mainly formed by the dissociation of \ce{N2H}, with a pathway such as
\begin{subequations}
\label{re:N_highP}
\begin{align}
\begin{split}
2 (\ce{ NH3 + H} &\rightarrow \ce{NH2 + H2 })\\
\ce{ NH2 + NH2} &\rightarrow \ce{N2H2 + H2 }\\
\ce{ N2H2 + H} &\rightarrow \ce{N2H + H2 }\\
\ce{ N2H + M} &\rightarrow \ce{N2 + H + M }\\
\ce{ H2 + M }&\rightarrow \ce{ 2H + M }\\
\hline \nonumber
\mbox{net} : \ce{ 2NH3} &\rightarrow \ce{N2 + 3H2 }
\end{split}
\end{align}
\end{subequations}
where the second reaction (\ref{R11}) is the RLS. As temperature increases, (\ref{R11}) is replaced by (\ref{R13}), or a channel through \ce{N2H3} and \ce{N2H4}  (\ref{R12}). This pathway is similar to that identified for HD 209458b by \cite{moses11}.

At low pressures (the parameter space occupied by (\ref{R14}) and (\ref{R15}) in Table \ref{tab: pathways_N2}), H$_2$ is attacked by the more abundant free atomic O and produces OH, which in turn forms NO with N via \ce{ N + OH } $\rightarrow$ \ce{ NO + H }. NO then react with N or NH$_2$, depending on the temperature, to produce N$_2$. This step involves forming the N$\equiv$N bond and is usually the RLS. The pathway becomes
\begin{subequations}
\label{re:N_highP}
\begin{align}
\begin{split}
2(\ce{ NH3 + H} &\rightarrow \ce{NH2 + H2 })\\
2(\ce{ NH2 + H} &\rightarrow \ce{NH + H2 })\\
2(\ce{ NH + H} &\rightarrow \ce{N + H2 })\\
\ce{ N + NO }&\rightarrow \ce{ N2 + O }\\
\ce{ O + H2 }&\rightarrow \ce{ OH + H }\\
\ce{ N + OH }&\rightarrow \ce{ NO + H }\\
2(\ce{ H2 + M }&\rightarrow \ce{ 2H + M })\\
\hline \nonumber
\mbox{net} : \ce{ 2NH3} &\rightarrow \ce{N2 + 3H2 }
\end{split}
\end{align}
\end{subequations}
where (\ref{R15}) is the RLS.

\begin{comment}
NH$_3$-HCN {\bf interconversion schemes can be generally divided into \ce{CH4} and CO channels. At high pressures and low temperatures,  where \ce{CH4} is the dominant carbon species, NH$_3$ reacts with \ce{CH4} to form HCN. At low pressures and high temperatures, CO becomes the dominant carbon species and provides the carbon to make HCN.}

In the \ce{CH4} channel (the parameter space occupied by (\ref{R16}) in Table \ref{tab: pathways_HCN}), the dominant pathway is
\begin{subequations}
\label{re:HCN_highP}
\begin{align}
\begin{split}
\ce{ NH3 + H} &\rightarrow \ce{NH2 + H2 + M}\\
\ce{ NH2 + CH3 + M} &\rightarrow \ce{CH3NH2}\\
\ce{ CH3NH2 + H} &\rightarrow \ce{CH2NH2 + H2}\\
\ce{ CH3NH2 + H} &\rightarrow \ce{CH2NH + H2}\\
\ce{ CH2NH + H }&\rightarrow \ce{ H2CN + H2 }\\
\ce{ H2CN + M }&\rightarrow \ce{ HCN + H + M }\\
\ce{ CH4 + H }&\rightarrow \ce{ CH3 + H2 }\\
2(\ce{ H2 + M }&\rightarrow \ce{ 2H + M })\\
\hline \nonumber
\mbox{net} : \ce{ NH3 + CH4} &\rightarrow \ce{HCN + 3H2}
\end{split}
\end{align}
\end{subequations}
where (\ref{R16}) is the RLS.

In the CO channel (the rest of the parameter space) in Table \ref{tab: pathways_HCN}), the dominant pathway becomes
\begin{subequations}
\label{re:HCN_highP}
\begin{align}
\begin{split}
\ce{ NH3 + H} &\rightarrow \ce{NH2 + H2 }\\
\ce{ NH2 + CO} &\rightarrow \ce{HNCO + H}\\
\ce{ HNCO + H} &\rightarrow \ce{HCN + OH}\\
\ce{ OH + H2} &\rightarrow \ce{H2O + H}\\
\hline \nonumber
\mbox{net} : \ce{ NH3 + CO} &\rightarrow \ce{HCN + H2O}
\end{split}
\end{align}
\end{subequations}
where (\ref{R17}) is the RLS.
\end{comment}

\begin{table} [!h]
\begin{center}
\caption{\ce{NH3} $\leftrightarrow$ \ce{N2} rate-limiting reactions}
\label{tab: pathways_N2}
\begin{tabular}{cccccc}
\hline
\textbf{T (K)} & 500 &  1000  &  1500  &  2000  &  2500  \\
\hline
\textbf{P (bar)} &&&&&\\
10$^{-4}$ & R11 & R14 & R15 & R15 & R15\\
10$^{-3}$ & R11 & R14 & R13 & R15 & R15\\
10$^{-2}$ & R11 & R13 & R13 & R15 & R15\\
10$^{-1}$ & R11 & R12 & R13 & R13 & R15\\
1         & R12 & R12 & R13 & R13 & R13\\
10        & R11 & R12 & R13 & R13 & R13\\
100       & R11 & R12 & R16 & R13 & R13\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{comment}
\begin{table} [!h]
\begin{center}
\caption{\ce{NH3} $\leftrightarrow$ \ce{HCN} rate-limiting reactions}
\label{tab: pathways_HCN}
\begin{tabular}{cccccc}
\hline
\textbf{T (K)} & 500 &  1000  &  1500  &  2000  &  2500  \\
\hline
\textbf{P (bar)} &&&&&\\
10$^{-4}$ & R17 & R17 & R17 & R18 & R18\\
10$^{-3}$ & R17 & R17 & R17 & R18 & R18\\
10$^{-2}$ & R16 & R17 & R17 & R17 & R18\\
10$^{-1}$ & R16 & R17 & R17 & R17 & R18\\
1              & R16 & R16 & R17 & R17 & R17\\
10            & R16 & R16 & R16 & R17 & R17\\
100          & R16 & R16 & R16 & R16 & R19\\
\hline
\end{tabular}
\end{center}
\end{table}
\end{comment}

\section{Computing chemical timescales}
\label{sec:chemtime}

The full expressions for the chemical timescales, including those involving methane, carbon monoxide, water, ammonia and molecular nitrogen, are stated in Appendix \ref{app:f_tau}.  In the following, we explain the reasoning behind their construction.

\subsection{Revisiting CH$_4$-CO interconversion}

\begin{table
\begin{center}
\caption{\ce{CH4} $\leftrightarrow$ \ce{CO} rate-limiting reactions (solar abundance)}
\label{tab: pathways}
\begin{tabular}{cccccc}
\hline
\textbf{T (K)} & 500 &  1000  &  1500  &  2000  &  2500  \\
\hline
\textbf{P (bar)} &&&&&\\
10$^{-4}$ & R9 & R2 & R3 & R5 & R6 \\
10$^{-3}$ & R9 & R2 & R3 & R4 & R5 \\
10$^{-2}$ & R9 & R2 & R2 & R4 & R5 \\
10$^{-1}$ & R9 & R1 & R2 & R3 & R4 \\
1         & R9 & R1 & R2 & R3 & R3 \\
10        & R9 & R1 & R1 & R3 & R3 \\
100       & R9 & R1 & R1 & R7 & R7 \\

\hline
\end{tabular}
\end{center}
\end{table}

\begin{table
\begin{center}
\caption{\ce{CH4} $\leftrightarrow$ \ce{CO} rate-limiting reactions (C/O = 2)}
\label{tab: pathways_CtoO2}
\begin{tabular}{cccccc}
\hline
\textbf{T (K)} & 500 &  1000  &  1500  &  2000  &  2500  \\
\hline
\textbf{P (bar)} &&&&&\\
10$^{-4}$ & R9 & R2 & R8  & R8 & R5 \\
10$^{-3}$ & R9 & R2 & R8  & R8 & R8 \\
10$^{-2}$ & R9 & R2 & R10 & R8 & R8 \\
10$^{-1}$ & R9 & R1 & R2  & R8 & R8 \\
1         & R9 & R1 & R2  & R3 & R8 \\
10        & R9 & R1 & R1  & R3 & R8 \\
100       & R9 & R1 & R7  & R7 & R7 \\
\hline
\end{tabular}
\end{center}
\end{table}

The study of interconversion between methane and carbon monoxide has a long and rich history. In the current study, we focus on identifying the pathways in reducing atmospheres, while the same steps can be applied to other types of atmospheres. In this subsection, our goal is to provide an analytical expression for computing the timescale associated with CH$_4$-CO interconversion as a function of the reactions (\ref{R1}) to (\ref{R10}).  Table \ref{tab: pathways} shows the RLSs for solar metallicity, while Table \ref{tab: pathways_CtoO2} shows them for $\mbox{C/O}=2$.  Reactions (\ref{R8}) and (\ref{R10}) are only relevant for $\mbox{C/O}=2$, as previously mentioned.

Under differing conditions of temperature and pressure, the various RLSs can either collaborate or compete with one another.  We find that a useful analogy for understanding the pathways is to visualize them as the resistors in an electrical circuit---built in series or in parallel.  Using such an analogy, we can construct an analytical expression for the chemical timescale that consists of a network of reactions working either in series or in parallel. The upper diagram of Figure \ref{fig:circuit} visualizes how such an analogous electrical circuit would look like for CH$_4$-CO interconversion. We identify the series and parallel RLSs and group the RLSs that operate in similar temperatures and pressures together. For example, (\ref{R2}), (\ref{R3}) are in series but (\ref{R2}) and (\ref{R4}) are in parallel.  Depending on the temperature and pressure, either (\ref{R1}), (\ref{R9}), or the group of (\ref{R2}), (\ref{R3}), and (\ref{R4}) is in control.

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width=0.75\columnwidth]{f6a.pdf}
\hspace*{-0.3cm}\includegraphics[width=0.67\columnwidth]{f6b.pdf}
\end{center}
\vspace{-0.2in}
\caption{The effective ``electronic circuit" of the major chemical pathways between methane and carbon monoxide (upper) for Figure \ref{fig:scheme} and between ammonia and molecular nitrogen (lower) for Figure \ref{fig:N_scheme}. The dashed rectangles in blue, yellow, orange, and purple group the RLSs according to their operating temperatures and pressures.}
\label{fig:circuit}
\end{figure}

Mathematically, a pair of reactions in parallel can be expressed as an operation that takes the maximum of the two rate coefficients.  For a pair of reactions in series, the operation instead takes the minimum of the two rate coefficients. The groups that operate in particular temperatures and pressures are then added together for simplicity. In this way, the series of relationships between the reactions can be expressed as
\begin{equation}
\begin{split}
\tau_{\rm CH_4} &= \frac{[\mbox{CH}_4]}{\mbox{r1} + \max{\left( \min{\left( \mbox{r2,r3} \right)}, \mbox{r4} \right)} + \mbox{r9} + \min{\left( \mbox{r5,r6} \right)} }\\
& + \tau_{\rm H_2}\times \frac{3 [{\rm CO}]}{[{\rm H_2}]},
\end{split}
\end{equation}
where [X] represents the equilibrium number density of species X in cm$^{-3}$, rx stands for the reaction rate of (Rx), and the factor $\frac{3 [{\rm CO}]}{[{\rm H_2}]}$ stems from the amount of hydrogen that participates in the \ce{CH4}-CO interconversion (three \ce{H2} for every CO according to the net reaction). The second term is usually orders of magnitude smaller than the first term except at high temperatures and low pressures, where the dissociation and recombination between H and H$_2$ become important and the relatively slower conversion of hydrogen starts to bottleneck the process. We demonstrate how methane is controlled by hydrogen in Figure \ref{fig:sensitivity} where we manually vary the rate of H-H$_2$ dissociation/recombination.

We have also replaced (\ref{R7}) with (\ref{R1}) for simplicity because the rates of both reactions are very similar at high pressures. The same formula is applied to $\tau_{\rm CO}$ except one needs to replace the numerator with [CO], since the interconversion goes both ways.

\subsection{Water}

H$_2$O reacts efficiently with atomic hydrogen via
\begin{reactions}\label{R18}
\ce{H2O + H -> OH + H2}
\end{reactions}
in a hydrogen-rich atmosphere. Being the major oxygen carrier, water prevailingly participates in various reaction pathways, e.g. CO $\leftrightarrow$ CO$_2$ and H$_2$ $\leftrightarrow$ 2H. For solar metallicity, carbon is the richest heavy element next to oxygen. We find the timescale of water is effectively determined by the interconversion rate of CH$_4$-CO,
\begin{equation} \label{eq:tau_f}
\begin{split}
\tau_{\ce{H2O}} &= \frac{[\mbox{H}_2\mbox{O}]}{\mbox{r1} + \max{\left( \min{\left( \mbox{r2,r3} \right)}, \mbox{r4} \right)} + \mbox{r9} + \min{\left( \mbox{r5,r6} \right)} }\\
& + \tau_{\rm H_2}\times \frac{3 [{\rm CO}]}{[{\rm H_2}]}.
\end{split}
\end{equation}
\subsection{Carbon dioxide}

Carbon dioxide is produced through the relatively fast scheme \citep{lly10,moses11}
\begin{equation*}
\begin{split}
\ce{H2O + H} & \rightarrow \ce{ OH + H2 }\\
\ce{CO + OH} & \rightarrow \ce{ CO2 +H }\\
\hline \nonumber
\mbox{net} : \ce{CO + H2O} & \rightarrow \ce{CO2 + H2}
\end{split}
\end{equation*}
The chemical timescale is simply
\begin{equation}\label{eq:tau_co2}
\tau_{\mbox{CO}_2} = \frac{[\mbox{CO}_2]}{k_{\mbox{CO}_2}[\mbox{CO}][\mbox{OH}]}
\end{equation}
where $k_{\mbox{CO}_2}$ is the rate coefficient for \ce{CO + OH -> CO2 +H }.  Due to the fast conversion, CO$_2$ still maintains pseudo-equilibrium with CO and H$_2$O after the latter two are quenched, before CO$_2$ reaches its own quench point (see the discussion in section 3.1 of \citealt{moses11}). Owing to this coupling, instead of relaxing CO$_2$ to its equilibrium value, we find it correct to relax CO$_2$ toward the pseudo-equilibrium value determined by the (possibly quenched) CO and H$_2$O, as expressed in equation (\ref{eq:co2_pseudo}), similar to the treatment in equation (43) of \cite{zm14}.

\subsection{Ammonia and molecular nitrogen}

The chemical timescale of ammonia is approximately determined by the \ce{NH3}-\ce{N2} conversion. Omitting HCN, we group the RLSs with similar operating temperatures and pressures and construct the effective electrical circuit in the lower diagram of Figure \ref{fig:circuit}.

For ammonia production, we have
\begin{equation}
\tau_{\rm NH_3} = \frac{1}{2} \left( \frac{[\mbox{NH}_3]}{ \max{\left(\mbox{r11,r12}\right)} + \mbox{r13} + \mbox{r14}   + \mbox{r15}} +\tau_{\rm H_2}\times \frac{3 [{\rm N2}]}{[{\rm H_2}] } \right)
\end{equation}
where the factor $\frac{3 [{\rm N2}]}{[{\rm H_2}]}$ is again the amount of hydrogen that participates in the \ce{NH3}-\ce{N2} interconversion, as in the \ce{CH4}-CO interconversion limited by H-H$_2$ interconversion at high temperatures and low pressures, and the factor of 1/2 comes from the fact that the net reaction converts two NH$_3$ molecules to one N$_2$ molecule.

Following the same steps, the chemical timescale of molecular nitrogen is expressed as
\begin{equation}
\tau_{\rm N2} = \frac{[\mbox{N}_2]}{ \max{\left(\mbox{r11,r12}\right)} + \mbox{r13} + \mbox{r14}   + \mbox{r15}} +\tau_{\rm H_2}\times \frac{3 [{\rm N2}]}{[{\rm H_2}] }
\end{equation}

\begin{comment}
\begin{equation*}
\tau_{\rm NH_3-N_2}^\ast = \frac{1}{2} \frac{[\mbox{NH}_3]}{\mbox{R11} + \mbox{R12} + \max{\left(\mbox{R13,R14}\right)}+\mbox{R15}},
\end{equation*}
\begin{equation*}
\tau_{\rm NH_3-HCN}^\ast = \frac{[\mbox{NH}_3]}{\mbox{R16} + \mbox{R17} + \mbox{R18} + \mbox{R19} + \mbox{R20}},
\end{equation*}
\begin{equation}
\tau_{\rm NH_3}^\ast = \frac{\tau_{\rm NH_3-N_2}^\ast \times [\ce{N2}] + \tau_{\rm NH_3-HCN}^\ast \times [\mbox{HCN}]}{[\ce{N2}] + [\mbox{HCN}]}
\end{equation}
where the factor of 1/2 comes from the fact that the net reaction converts two NH$_3$ molecules to one N$_2$ molecule.  The process is again limited by H-H$_2$ interconversion,
\begin{equation}
\tau_{\rm H} = \frac{[\mbox{H}]}{\mbox{H}_2-\mbox{H interconversion rate}},
\end{equation}
such that the final chemical timescale is
\begin{equation}
\tau_{\rm NH_3} = \max{\left(\tau_{\rm NH_3}^\ast,\tau_{\rm H}\right)}.
\end{equation}
\end{comment}

\begin{comment}
\begin{equation*}
\tau_{\rm N_2-NH_3}^\ast = \frac{[\mbox{N}_2]}{\mbox{R11} + \mbox{R12} + \max{\left(\mbox{R13,R14}\right)}+\mbox{R15}},
\end{equation*}
\begin{equation*}
\tau_{\rm N_2-HCN}^\ast = \frac{[\mbox{N}_2]}{\mbox{R16} + \mbox{R17} + \mbox{R18} + \mbox{R19} + \mbox{R20}},
\end{equation*}
\begin{equation*}
\tau_{\rm N_2}^\ast = \frac{\tau_{\rm N_2-NH_3}^\ast \times [\ce{NH3}] + \tau_{\rm N_2-HCN}^\ast \times [\mbox{HCN}]}{[\ce{NH3}] + [\mbox{HCN}]}
\end{equation*}
\begin{equation}
\tau_{\rm N_2} = \max{\left(\tau_{\rm N_2}^\ast,\tau_{\rm H}\right)}.
\end{equation}
\end{comment}

\subsection{The effects of metallicity}

As the metallicity is varied from $10^{-2} \times$ to $1000 \times$ of the solar values, we find that the chemical pathways and RLSs discussed in section \ref{sec:rls} do not change. Therefore, the formulae for the timescales remain the same. An exception is for the NH$_3$-N$_2$ pathways as the metallicity approaches $100 \times$ solar. (\ref{R14}) and (\ref{R15}) occupy more of the temperature-pressure parameter space because of the richness of oxygen.  We also confirm that $\tau_{\rm CO}$ is almost independent of metallicity and $\tau_{\rm CH_4}$ is inversely proportional to metallicity, as found by \cite{vc12}.

Once the metallicity increases beyond $1000 \times$ solar, the atmosphere ceases to be H$_2$-dominated and the main constituents become carbon dioxide or molecular oxygen \citep{hu14}.  For example, at $10^4\times$ solar metallicity, CH$_4$ becomes scarce simply because of the lack of hydrogen for its formation.  In this scenario, CO-CO$_2$ interconversion becomes the main quenching process and follows the same pathway as for solar metallicity with the timescale still given by equation (\ref{eq:tau_co2}).  The pathways involving the nitrogen species become completely altered.  Generally, the reactions of NH$_{\rm x}$ with H$_2$O become important in controlling NH$_3$-N$_2$ interconversion.

\subsection{The effects of C/O}

The carbon-to-oxygen ratio (C/O) is a crucial factor in controlling the atmospheric chemistry and thermal structure \citep{madhu12,moses13a,venot15,rocc16}. Equilibrium chemistry is sensitive to C/O and undergoes a qualitative transition at $\mbox{C/O} = 1$. We explore C/O values ranging from 0.1 to 2. The reason to limit ourselves at $\mbox{C/O} = 2$ is that C/O much larger than unity is considered unlikely as the surplus carbon tends to condense and form graphite \citep{moses13b}. Comparing Table \ref{tab: pathways} and Table \ref{tab: pathways_CtoO2}, the major change as C/O exceeds unity is carbon takes the route through C$_2$H$_2$ at high temperature.  The typical pathway in a hot, carbon-rich atmosphere is
\begin{subequations}
\begin{align}
\begin{split}
\ce{ CH4 + H} &\rightarrow \ce{ CH3 + H2 } \\
\ce{ CH3 + CH2} &\rightarrow \ce{ C2H4 + H} \\
\ce{ C2H4 + H} &\rightarrow \ce{ C2H3 + H2} \\
\ce{ C2H3 + M} &\rightarrow \ce{ C2H2 + H + M} \\
\ce{ C2H2 + O} &\rightarrow \ce{ CH2 + CO} \\
\ce{ OH + H} &\rightarrow \ce{ O + H2 } \\
\ce{ H + H2O }&\rightarrow \ce{ OH + H2 } \\
\ce{ H2 + M}&\rightarrow \ce{ 2H + M } \\
\hline \nonumber
\mbox{net} : \ce{ CH4 + H2O} &\rightarrow \ce{CO + 3H2 }
\end{split}
\end{align}
\end{subequations}
where \ce{ C2H2 + O} $\rightarrow$ \ce{CH2 + CO } (\ref{R8}) is the RLS in the above pathway. The corresponding timescale for C/O between 1 and 2 is
\begin{equation}
\tau_{\rm CH_4} = \frac{[\mbox{CH}_4]}{\mbox{r1} + \min{\left( \mbox{r2,r3} \right)} + \mbox{r9} + \max{\left( \mbox{r8,r10} \right)}},
\end{equation}
In this scheme, CH$_4$-CO conversion is no longer limited by hydrogen dissociation.

\section{Comparison of chemical networks using pathway analysis}
\label{sec:pathway}

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width= 0.3\columnwidth]{f7a.pdf}
\includegraphics[width= 0.3\columnwidth]{f7b.pdf}
\includegraphics[width= 0.3\columnwidth]{f7c.pdf}
\end{center}
\vspace{-0.2in}
\caption{Examples of CH$_4$-CO pathway analysis at $T = 2000$ K and $P = 0.1$ bar with the chemical network from VULCAN (left), \cite{moses11} (middle), and \cite{venot12} (right). Thicker lines represent faster reaction rates (denoted by the first-row numbers shown in in cm$^{-3}$s$^{-1}$ and the percentage of contribution to the interconversion rate is also provided) and the red lines are the rate-limiting steps.}
\label{fig:path_tool}
\end{figure}

We recognize that the chemical pathways taken depend entirely on the network one is using. With a different network, the rate-limiting chemical reactions need to be re-identified. The expressions for chemical timescales can be worked out following the same steps in \S\ref{sec:chemtime}. Here, we compare our network with two others that have included high-temperature chemical kinetics and been applied to hot exoplanets.  We first compare our network with that of \cite{moses11}; our two networks are naturally similar, because we have taken the values of certain rate coefficients from that study.  We then compare our network with that of \cite{venot12}.  In Figure \ref{fig:path_tool}, we use our pathway analysis tool to compare the route taken by each network (Since the reaction rates are calculated with equilibrium composition, the forward rate equals to the reverse rate, there is no directionality in the pathways, i.e. CH$_4$ $\rightarrow$ CO takes the same pathway as CO $\rightarrow$ CH$_4$).

At solar abundance and temperatures less than 2000 K, the network of \cite{moses11} shows almost exactly the same CH$_4$-CO pathways as ours. Due to a different choice of the rate coefficients associated with (\ref{R2}), part of the parameter space occupied by (\ref{R3}) is replaced by (\ref{R2}) in their network for $T \gtrsim 2000$ K and $P \sim 10$ bar.  (\ref{R7}) also extends to lower pressures in their network.  At high temperatures and low pressures in scheme (A), their network experiences the same dehydrogenation process, but instead of (\ref{R4}) and (\ref{R5}), their network chooses a faster path through water: \ce{CH + H2O} $\rightarrow$ {H2CO + H} (which is not included in our network).  Yet, in this regime, the timescale is limited by hydrogen dissociation and yields the same timescale of CH$_4$-CO interconversion as we do.  At $\mbox{C/O} = 2$, carbon forms abundant \ce{C2H2} and then gets oxidized to CO in a similar way, but except via (\ref{R10}), it takes \ce{C2H2 + OH} $\rightarrow$ \ce{H2CCO + H} or \ce{C2H2 + O} $\rightarrow$ {HCCO + H} in their network. \ce{H2CCO} or \ce{HCCO} then proceeds to be split into CO by H. In general, our network is consistent with that of \cite{moses11}, as suggested by the comparison in \cite{tsai17}.

The overall CH$_4$--CO timescale in \cite{venot12} is shorter than ours and \cite{moses11}.  We find the two key reactions that make the RLSs and essentially the timescales different are (\ref{R9}) and
\begin{reactions}\label{V1}
\ce{CH2OH + M} \rightarrow \ce{H2CO + H + M}.
\end{reactions}
\cite{venot12} includes faster rate coefficients for both (\ref{R9}) and (\ref{V1}).  Their rate coefficient for (\ref{R9}) is based on the work of \cite{hidaka}, which has been suggested as overestimating the rate; see discussion in \cite{vc12} and \cite{moses14}. This rate coefficient is significantly higher ($\sim 10$ orders of magnitudes) than the ab initio calculation in \cite{moses11}, which is also used in our network.

At lower temperatures (T $\lesssim$ 1000 K), \cite{venot12} takes the same pathway through (\ref{R9}), forming \ce{CH3OH} from \ce{CH3} and \ce{H2O}. However, with a faster rate it never bottlenecks the pathway and controls the timescale. Their RLSs are instead the reactions involving forming or destroying \ce{H2CO}, e.g. \ce{CH3O + M} $\rightarrow$ \ce{H2CO + H + M} and \ce{H2CO + H} $\rightarrow$ \ce{HCO + H2}. Similarly, due to the faster \ce{CH3}-\ce{CH3OH} channel via (\ref{R9}), for $1000 \mbox{ K} \lesssim T \lesssim 1500$ K, \cite{venot12} exhibits pathways close to ours, except that our (\ref{R1}) is replaced by \ce{CH2OH + H2} $\rightarrow$ \ce{CH3OH + H} as the RLS. At higher temperature where $T \gtrsim 1500$ K, the differences are mainly attributed to (\ref{V1}). \cite{venot12} uses the rate from \cite{gh86}, validated for 600--1000 K, while this work and \cite{moses11} use the rate from \cite{cribb92}, validated for 1900--2700 K. The former is about two orders of magnitude larger than the latter in this temperature range. Consquently, the fast \ce{CH2OH}-\ce{H2CO} interconversion in \cite{venot12} again never limits the pathway (e.g. the right pathway in Figure \ref{fig:path_tool}). Their RLS remains \ce{CH2OH + H2} $\rightarrow$ \ce{CH3OH + H} for high pressures and switches to \ce{CH3 + OH} $\rightarrow$ \ce{CH2OH + H} for low pressure.

In conclusion, our pathway analysis tool is useful in identifying the key reactions for a given network, which allows us to diagnose the divergent behaviors of different networks.  By isolating the rate coefficients of the key reactions involved, we hope to motivate future laboratory and/or theoretical studies that will hopefully resolve these discrepancies.

\section{Validation of chemical relaxation method}
\label{sec:validate}

\begin{figure
\begin{center}
\vspace{-0.1in}
\includegraphics[width=\columnwidth]{f8.eps}
\end{center}
\vspace{-0.2in}
\caption{Representative temperature-pressure profiles representing hot, warm, and cool atmospheres. Gray dashed curves show the boundaries where \ce{CH4}-CO and \ce{NH3}-\ce{N2} have equal abundances in chemical equilibrium.}
\label{fig:KTP}
\end{figure}

We are finally ready to validate the chemical relaxation method to a factor of several, having assembled the necessary ingredients.
\subsection{Setup}

The goal of any chemical calculation, either from full kinetics or any simplified method, is to provide the rate of change of every species locally. The applicability of the relaxation method should not depend on the format or complexity of atmospheric dynamics. Therefore, before including the chemical relaxation method in a GCM, we evaluate whether and to what extend the relaxation method can replace full chemical kinetics with a one-dimensional full kinetics model, where eddy diffusion ($K_{zz}$) is used to represent vertical mixing.  For $w$ (vertical velocity) $\sim 1$ km s$^{-1}$ and $H \sim 100$ km, we have $K_{zz} \lesssim w H \sim 10^{12}$ cm$^2$ s$^{-1}$.  To exaggerate the effects of vertical mixing, we explore a range of $K_{zz}$ values up to $10^{15}$ cm$^2$ s$^{-1}$.

We run the same atmospheric conditions with the relaxation method and \texttt{VULCAN}, a full kinetics model with a C-H-O network including 29 species with up to two carbon atoms and about 300 forward and reverse reactions \citep{tsai17}. For nitrogen chemistry, an updated N-C-H-O network is implemented including 53 species and about 600 forward and reverse reactions. The chemical equilibrium abundances are calculated using the \texttt{FastChem}\citep{fastchem} code. We perform our chemistry calculations over three temperature-pressure profiles constructed using the analytical formula in \cite{hml14} to represent cool, warm and hot atmospheres (Figure \ref{fig:KTP}). These profiles are meant to mimic GJ 1214b-, HD 189733b- and WASP-18b-like atmospheres. With the atmospheres we tested, the computational time for integrating one step using the chemical relaxation method is about 5 ms, compared to about 0.5 s using \texttt{VULCAN}. The computational speed can be increased by about 100 times with the chemical relaxation method.

\subsection{Hot atmospheres (WASP-18b-like)}

At solar metallicity and high temperatures, CO, H$_2$O, and N$_2$ are the dominant molecules in chemical equilibrium \citep{madhu12,moses11,ht16}.  In Figure \ref{fig:hot_valid}, it is therefore unsurprising that the mixing ratios of the first two molecules (not showing N$_2$) are insensitive, or nearly independent of, pressure.  For $K_{zz} \lesssim 10^{13}$ cm$^2$ s$^{-1}$, the mixing ratios of CO and H$_2$O essentially track their chemical-equilibrium values closely, deviating only with stronger mixing and/or lower pressures ($\lesssim 1$ mbar).

The mixing ratios of CH$_4$ and NH$_3$ exhibit a much larger range of values as they both drop off significantly with increasing altitude: 13 and 7 orders of magnitude for CH$_4$ and NH$_3$, respectively, over the range of pressure examined (0.1 mbar to 1 kbar).  Over these broad ranges, the chemical relaxation method performs fairly well, exhibiting an accuracy of within an order of magnitude for the most part. With strong vertical mixing, CH$_4$ and NH$_3$ begin to quench at about 10 bar and 100 bar, respectively. At about 1 mbar, hydrogen dissociation/recombination slows down the interconversion and sets the second quench level. If the effect of hydrogen dissociation is neglected, the estimated chemical timescale will be too short and the prediction of CH$_4$ and NH$_3$ will be too close to chemical equilibrium.

\subsection{Warm atmospheres (HD 189733b-like)}

Figure \ref{fig:warm_valid} shows the mixing ratios of CO, CO$_2$, CH$_4$, H$_2$O and NH$_3$ for a HD 189733b-like atmosphere.  The chemical relaxation method performs with an accuracy of better than a factor of 2 in most parts across a broad range of pressures and mixing ratio values, with \ce{NH3} being least accurate due to the error in estimating its timescale.  For comparison, we show the chemical-relaxation calculations performed using our implementation of the method of \cite{cs06} as dot-dashed curves.  Recall that \cite{cs06} uses essentially a shorter single chemical timescale of CO (tied to a single RLS), whereas the main goal of the present study is to use a set of rate-limiting chemical reactions depending on the temperature and pressure conditions to obtain more accurate timescales.

\cite{cs06} and \cite{ben18} only calculate CO using the relaxation method and relate \ce{CH4} (and other species) through mass balance, assuming that all the carbons are locked in either \ce{CH4} and CO, hence the mixing ratios $X_{\ce{CH4}}$ and $X_{\ce{CO}}$ are conserved:
\begin{equation}\label{eq:mass_balance}
X_{\ce{CH4}} + X_{\ce{CO}} = \ce{C}/(\ce{H2}+\ce{He})
\end{equation}
where \ce{C}/(\ce{H2}+\ce{He}) is the ratio of carbon atoms to molecular hydrogen and helium (i.e. the bulk gas), determined by solar metallicity. We emphasize that this mass balance relation is only valid when (1) the system is in or close to chemical equilibrium (2) temperature is not too high such that all hydrogen remains in molecular form (3) \ce{CH4} and CO have close abundances. The elemental abundance ($\ce{C}/(\ce{H2}+\ce{He})$ in this case) is a local property, and can be violated by disequilibrium processes\footnotemark[3]. We demonstrate this in the top right panel in Figure \ref{fig:warm_valid}, with the mixing ratios of \ce{CH4} calculated from equation (\ref{eq:mass_balance}) using $X_{\ce{CO}}$ obtained from chemical relaxation (the top left panel in Figure \ref{fig:warm_valid}) in dot-dashed curves. The mass balance approach overpredicts the quenching of \ce{CH4}. Furthermore, as CO is $\sim$3 orders of magnitude more abundant than \ce{CH4} in this case, which requires the estimation of CO to be as accurate as $\sim$3 decimal places for the mass balance approach to work. Unfortunately, this precision is not attainable with the relaxation method or any other kinetics models. Therefore, the mass balance approach can only be applied to a system in chemical equilibrium but not applicable to the relaxation method.

\footnotetext[3]{{For example, CO quenching alone increases the metallicity relative to that in chemical equilibrium, although the effect of changing the metallicity is usually small since
the quenched species are trace gases.}}

For CO$_2$, we show for comparison the calculation of relaxing CO$_2$ to the equilibrium abundance of CO2 (dot-dashed curves) versus to its pseudo-equilibrium value as determined by the quenched abundances of CO and H$_2$O (solid curves).  The difference in accuracy is substantial: $\sim 10\%$ versus an order of magnitude.  Neglecting this effect will lead to the inaccurate prediction that the mixing ratio of CO$_2$ is close to equilibrium.

For illustration, we show the mixing ratios of CH$_4$ when the metallicity is $100 \times$ solar, as well as those of H$_2$O and CO$_2$ when $\mbox{C/O}=2$.  In the latter case, H$_2$O loses its dominance to CH$_4$ and its mixing ratio becomes sensitive to the strength of vertical mixing.  In these cases, our more general treatment of chemical relaxation allows the accuracy of the method to remain about the same as for the solar-metallicity case.

\subsection{Cool atmospheres (GJ 1214b-like)}

Figure \ref{fig:cool_valid} shows the mixing ratios of CO, CO$_2$, CH$_4$ and N$_2$ for a GJ 1214b-like atmosphere.  In this range of temperatures, the chemical relaxation method is highly accurate ($\sim 10\%$). For CO, the difference from using the single-RLS timescale of \cite{cs06} increases significantly to a few orders of magnitudes. However, we note that photochemistry will potentially influence the abundances in these cooler atmospheres, which is not taken into account in this work.

\begin{figure*
\begin{center}
\vspace{-0.1in}
\includegraphics[width= 0.87\columnwidth]{f9a.eps}
\includegraphics[width= 0.87\columnwidth]{f9b.eps}
\includegraphics[width= 0.87\columnwidth]{f9c.eps}
\includegraphics[width= 0.87\columnwidth]{f9d.eps}
\end{center}
\caption{Mixing ratios of CO,CH$_4$, H$_2$O, and NH$_3$ in the hot atmosphere as displayed in Figure \ref{fig:KTP}. The results of the relaxation method (dashed) are compared to the full chemical kinetics (solid) for a range of vertical mixing strengths shown in various colors.  The mixing ratio in chemical equilibrium is shown as a dotted curve.}
\label{fig:hot_valid}
\end{figure*}

\begin{figure*}[!h]
\begin{center}
\vspace{-0.1in}
\includegraphics[width= 0.87\columnwidth]{f10a.eps}
\includegraphics[width= 0.87\columnwidth]{f10b.eps}
\includegraphics[width= 0.87\columnwidth]{f10c.eps}
\includegraphics[width= 0.87\columnwidth]{f10d.eps}
\includegraphics[width= 0.87\columnwidth]{f10e.eps}
\includegraphics[width= 0.87\columnwidth ]{f10f.eps}
\includegraphics[width= 0.87\columnwidth ]{f10g.eps}
\includegraphics[width= 0.87\columnwidth ]{f10h.eps}
\end{center}
\caption{Same as Figure \ref{fig:hot_valid} but for the warm atmosphere. For CO, the chemical-relaxation calculations adopting the timescale from \cite{cs06} are shown as dashed-dotted curves. For \ce{CH4}, the mass balance approach using the chemical-relaxation calculations of CO are shown in dashed-dotted curves. For CO$_2$, the chemical-relaxation calculations without considering the coupling to CO and \ce{H2O} are shown in dashed-dotted curves for comparison.}
\label{fig:warm_valid}
\end{figure*}

\begin{figure*
\begin{center}
\vspace{-0.1in}
\includegraphics[width= 0.87\columnwidth]{f11a.eps}
\includegraphics[width= 0.87\columnwidth]{f11b.eps}
\includegraphics[width= 0.87\columnwidth]{f11c.eps}
\includegraphics[width= 0.87\columnwidth]{f11d.eps}
\end{center}
\caption{Same as Figure \ref{fig:hot_valid} but for the cool atmosphere. For CO, the chemical-relaxation calculations adopting the timescale from \cite{cs06} are shown as dashed-dotted curves. For CO$_2$, the chemical-relaxation calculations without considering the coupling to CO and \ce{H2O} are shown in dashed-dotted curves for comparison.}
\label{fig:cool_valid}
\end{figure*}

\section{Summary and discussion}
\label{sec:summary}

\subsection{Summary}

Inspired by the pioneering work of \cite{cs06}, we have revisited the chemical relaxation method, which seeks to greatly enhance computational efficiency by replacing the network in a chemical kinetics calculation with a few independent source/sink terms that ``relax" towards chemical equilibrium on a prescribed timescale.  There is a precedent of using relaxation methods as a substitute for radiative transfer, where the timescale is then associated with radiative cooling (e.g., \citealt{hs94}).  The main lessons learned from our study are:
\begin{itemize}

\item The rate-limiting reaction that determines the chemical timescale depends on the temperature and pressure.  For \ce{CH4}-CO and \ce{NH3}-\ce{N2}-HCN interconversion, we show across a broad range in temperature and pressure (500--3000 K, 0.1 mbar to 1 kbar) that there are multiple rate-limiting reactions, and that the chemical timescale cannot be easily fitted by an Arrhenius-like function.

\item By comparing full chemical-kinetics to chemical-relaxation calculations in one dimension, we show that the latter is accurate to within an order of magnitude for WASP-18b-like atmospheres, $\sim$ a factor of 2 for HD 189733b-like atmospheres and $\sim 10\%$ for GJ 1214-b-like atmospheres.  Essentially, the chemical relaxation method is more accurate when the species either fully quench or retain chemical equilibrium. The discrepancies become larger when the species do not fully quench, because the behavior is more sensitive to the timescale in this situation. Overall, species at lower temperatures tend to fully quench (e.g., \citealt{moses16}) since the timescale of the main quenched species (CO and \ce{N2}) quickly increases with altitude (see Figure \ref{fig:tau_contours}). This bodes well as the currently characterizable atmospheres will become cooler as observational methods advance, but the effects of photochemistry need to be examined.

\item The relaxation method increases the computational speed by at least 100 times compared to running a full kinetics model. More importantly, the relaxation method allows decoupling from the chemical network. Only the species of interest need to be included, which will significantly ease the burden of adding numerous tracers to the dynamical core.

\end{itemize}

\subsection{Opportunities for future work}

There are ample opportunities for future work. We have recently finished and submitted the work of coupling our chemical relaxation method to our GCM \citep{mendonca16} and studying the interaction between atmospheric dynamics and chemistry in three dimensions.  It remains an open question if photochemistry may be reasonably approximated by chemical relaxation, since it will require a pre-calculated photochemical steady state for a given atmospheric condition and stellar flux. Generalizing chemical relaxation to work in the regime of atmospheres with Earth-like temperatures will be useful as we march towards the study of exo-climates similar to our own.

\acknowledgments
S.-M.T. and K.H. acknowledge partial financial support from the PlanetS National Center of Competence in Research (NCCR), the Center for Space and Habitability, the Swiss National Science Foundation and the Swiss-based MERAC Foundation, as well as useful conversations with Julie Moses and Paul Rimmer.

\software{\\
Python\footnotemark[4], SciPy\footnotemark[5], NumPy\footnotemark[6] \citep{numpy}, Matplotlib\footnotemark[7] \citep{matplotlib}}
\footnotetext[4]{\url{http://www.python.org}}
\footnotetext[5]{\url{http://scipy.org}}
\footnotetext[6]{\url{http://numpy.org}}
\footnotetext[7]{\url{http://matplotlib.org}}

redpajama_set_name: RedPajamaArXiv


From a business perspective, it makes sense why MTV has stretched back into its vault to revive shows like "Daria," "Made" and "The Real World," which it plans on shopping to outside networks.
When the teen-friendly network brought back "Jersey Shore" in April as "Jersey Shore Family Vacation," the surprisingly high ratings said it all: people wanted to watch this familiar show more than they'd wanted to watch other MTV shows from recent years.
As Murray explains, the evolution of television has created an insatiable appetite for content — and with a reboot comes a "pre-sold product" that's previously found success.
And that's not to mention the nostalgia factor which certainly draws people in — a big selling point for shows like "Murphy Brown," which will return to CBS this fall, and "Will & Grace," which drew in monster ratings this season for NBC.
There's likely something about Will and Grace's witty banter that conjures some sort of feeling of the "good old days" among viewers.
"(Reboots take) us back maybe in a time that feels particularly dark for a lot of people … back to a perhaps less seemingly complex time," said Murray.
But right now we're facing nostalgia overload. In addition to the aforementioned shows, let's not forget about "Charmed," "The Jetsons," "Miami Vice," "The Munsters," "Animaniacs," "American Idol," "Magnum P.I.," "Dynasty," "Muppet Babies," "One Day at a Time," "Party of Five," "The X-Files," "Arrested Development," "Roswell," "She-Ra," "Ducktales," "Magic School Bus," "Gilmore Girls," "Mad About You," "Trading Spaces" and "Twin Peaks," which have all been or are in the process of being rebooted in recent years.
Sometimes it's best to let a good thing die. Let your favorite shows end as they're meant to end, because dragging something out past its time could likely tarnish whatever legacy it left behind.
Take a look at "Roseanne." Yes, the show's premiere was watched by nearly 1 in 10 Americans, and yes, it was the highest-rated series of the broadcast season. But that's not what people will remember about the rebooted sitcom. No, what they'll remember is that it was canceled after nine episodes because of its star's late-night Twitter habits.
A Paramount Network reboot of the 1988 black comedy "Heathers" was scrapped, too — for now, anyway – due to its focus on high school violence.
Let's also not forget that "24: Legacy" was canceled after just one season.
If the current TV landscape was lacking, the relentless batches of reboots would at least be understandable.
But we are living through what critics call "peak TV." Especially with the rise of streaming platforms like Netflix and Hulu, where viewers can find just about anything they're looking for. Sitcoms like "Black-ish," "Insecure" and "Unbreakable Kimmy Schmidt" are fresh, funny and best of all, totally original.
Dramas, too, are having a moment, from shows like "Westworld" to "The Handmaid's Tale," which, in addition to presenting new stories that haven't been endlessly rehashed, are genuinely powerful and thought-provoking — TV shows we haven't seen before, over and over again.
But hey, bring on that "Daria" reboot.


text:
Cord Project Launches Voice Messaging App That Borrows The Good Parts Of Texting
Anthony Ha	@anthonyha / 5 years
With the Cord Project, a pair of former Googlers are aiming to encourage people to use their smartphones for something a little unusual — talking to other people. You know, with your voice.
Now, the obvious question here is, "Uh, can't I already do that with my phone? Isn't that, like, the whole point?"
But literally right before I met up with Cord Project co-founders Thomas Gayno (CEO) and Jeff Baxter (designer), I had a moment that handily encapsulated some of the reasons behind the app. I was standing outside their office, shivering (it was a winter night in Brooklyn), struggling to text them (through gloves!) and let them know that I'd arrived. Then I remembered that I had this amazing device that could actually communicate instantaneously, and required only minimal use from my hands — why the heck was I bothering with text?
Now, my avoidance of phone calls has probably gotten more than a little out of hand, but people text a lot, is what I'm getting at. And while voice has been incorporated into other messaging apps, it's usually shoehorned in there with text and/or photos, rather than creating a new or unique interface. (For example, Gayno said that when people send multiple voice messages in WhatsApp, you end up with "this thread of play buttons," which isn't very appealing.)
And plain old talking still has plenty of advantages — not just in situations where you really want to get ahold of someone so they can let you into their office, but also, as Gayno argued, when a little more emotion is called for.
So The Cord Project's hope is to take the things that are good about texting (like the fact that you don't have to answer right away, and you don't have to listen to minute-long voicemails) and bring them back to voice messaging.
"You get the emotion and the nuance you get with a call with voice, but you also get the speed and asynchronous communication that you get with text," Baxter said. "It's this great middle ground that no one's nailed yet."
Gayno and Baxter gave me a quick demo of the app, which is called Cord, and I've been trying it out myself over the weekend (partly because other TC team members have started using it, too). It's a pretty simple interface — you see a homescreen with images of people you've talked to recently, and if you tap on those icons, you can listen to their most recent message. You can also hit the microphone button to record a message of your own — the message has to be 12 seconds or or less, and there's a little indicator that shows how much time you have left.
Other features include the ability to save your favorite messages, to send a message to multiple recipients, and to bring up your full phone book of contacts.
As soon as the demo ended, I brought up features that I'd want to add, like the ability to block other users, create groups, and search through my favorite voice messages. Apparently that's all on the road map, though some of the items are probably closer to fruition than others — Baxter said blocking will be "the very next thing we implement." And the long-term goal is also to move beyond smartphones to other devices, like smartwatches and other wearables. (After all, when Gayno and Baxter worked together at Google's Creative Lab, a lot of their work involved Google Glass.)
The Cord Project raised $1.8 million in seed funding led by Lerer Hippeau Ventures and Metamorphic Ventures, with participation from Google Ventures, Greycroft, Dave Morin's Slow Ventures, and Gary Vaynerchuk. Gayno and Baxter touched on a number of different possibilities for making money, like selling stickers or offering a version for team communication.
The company actually launched an iOS version of the app back in September, although Baxter and Gayno argued it was really just a test version — they consider today, with an improved iOS app and their first version on Android, as their true launch.
By the way, the Cord Project team also released an app called Shhout, with what Baxter called "anonymish" voice messaging. (It's not completely anonymous because, well, your voice isn't actually disguised.) Basically, it was an experiment with public messaging — Gayno said that while the company focus is on Cord itself, it may do other "little spinoff" apps in the future, with the aim of testing ideas that aren't quite ready for the main product.
"We come from this culture at Creative Lab and Google of really experimenting and trying things even if they might seem weird," Baxter said. "We wanted to continue that similar spirit."

redpajama_set_name: RedPajamaCommonCrawl


In NYC, they don't allow drinking alcoholic beverages in the parks.
Both allow and let can be used with the same meaning, which is permit. However the way that we use these words is different. For today's English lesson let's have a look at how we can use allow and let in everyday English.
They don't allow smoking on airplanes anymore.
New York City doesn't allow drinking alcoholic beverages in public.
Many colleges don't allow eating and drinking in the classroom.
They don't allow passengers to smoke on airplanes anymore.
New York City doesn't allow anyone to drink alcoholic beverages in public.
Many colleges don't allow students to eat or drink in the classroom.
Passengers are not allowed to smoke on airplanes anymore.
In New York City, you are not allowed to drink alcoholic beverages in public.
In many colleges students are not allowed to eat or drink in the classroom.
They don't let passengers smoke on airplanes anymore.
New York City doesn't let people drink alcoholic beverages in public.
Many colleges don't let students eat or drink in the classroom.


text:
Clairy Browne & The Bangin' Rackettes: Watch The Group Now On NBC TV'S "Last Call With Carson Daly;" Debut Album 'Baby Caught The Bus' Out May 21 On Vanguard Records
Posted on May 6, 2013 by MSOPR
DT: MAY 6, 2013
FR: LIBBY COFFEY/MSO
WATCH CLAIRY BROWNE & THE BANGIN' RACKETTES
INTERVIEW AND PERFORMANCE
ON NBC TV'S "LAST CALL WITH CARSON DALY"
DEBUT ALBUM 'BABY CAUGHT THE BUS' TO BE RELEASED
MAY 21 ON VANGUARD RECORDS
Watch Clairy Browne & the Bangin' Rackettes interview and performance on NBC TV's "Last Call With Carson Daly."
http://www.nbc.com/last-call-with-carson-daly/video/clairy-browne-and-the-bangin-rackettes/n36372
Already breakout stars in their native Australia, Clairy Browne & The Bangin' Rackettes–who fuse soul, blues, doo-wop and R&B with Clairy's powerful vocals and a pop-twist— have signed with U.S. label Vanguard Records who will release their debut album BABY CAUGHT THE BUS on May 21. In its first week at radio, their first single "Love Letter" was the #1 Most Added at Triple A Radio with airplay at Alternative stations across America as well. Expect the group that the New York Music Daily described as "the great long-lost collaboration between David Lynch and John Waters" (April 1, 2013) to appear on a forthcoming episode of NPR's "Weekend Edition" and to announce a U.S. summer tour soon.
CLAIRY BROWNE & THE BANGIN' RACKETTES and their debut album BABY CAUGHT THE BUS are gaining strong praise from the media. Bust magazine praised the album as "Big, soulful and undeniably retro" (March 2013). The LA Record hailed in a live review, "The entire band gave us a show full of sultry ballads and clap-a-long jams. The new single 'Love Letter' was a highlight as was the killer encores with great sax solos by Darcy McMulty. Browne won me over with the medley of 'The Boy Is Mine/She Plays Up to You' which momentarily broke the 60s spell with a 90s guilty pleasure" (4/11/13). The New York Music Daily added this re the band's explosive show: "Watching this band was like discovering the great long-lost collaboration between David Lynch and John Waters." (April 1, 2013). The Patriot Ledger also lauded the young group's "skillful blend of classic rhythm and blues, soulful jazz, blues and even some doo-wop vocal parts" (March 28, 2013).
http://clairybrowne.com
http://www.vanguardrecords.com/
https://www.facebook.com/ClairyBrowneandtheBanginRackettes
https://twitter.com/clairybrowne
https://soundcloud.com/clairybrowne
MSO Press releases & Assets:
http://www.msopr.com/n/client-roster/clairy-browne-and-the-bangin-39-rackette
Click here for BABY CAUGHT THE BUS high res cover art: http://msopr.com/files/clairybrowne.babycaughtthebus.jpg
Pictured above: Clairy Browne & The Bangin' Rackettes. Download photo: http://msopr.com/files/clairybrowne_pr1.JPG.
Official video for the first single "Love Letter:" http://youtu.be/irNtyaNHq5A
Bust magazine feature: http://msopr.com/files/bust_april_may_2013.pdf
About Clairy Browne & the Bangin' Rackettes:
Formed in 2009 over a mutual love of early rhythm n' blues records and rock n' roll, singer-songwriter Clairy Browne and her harmonizing group of rogue musicians The Bangin' Rackettes began rehearsing in an abandoned coffin factory in the suburbs of Melbourne. Together the nine-piece band has come a long way thrilling both festival and club crowds alike with their young, modern take on classic soul, rhythm and blues, jazz and doo-wop which plays a large part in their broad multi-generational appeal. Clairy Browne & the Bangin' Rackettes—Clairy Browne (vocals), Camilla (vocals), Loretta Miller (vocals), Ruby Jones (vocals), Jules "Crazy Legs" Pascoe (bass guitar), Darcy McNulty (baritone sax), Gabriel Strango (Piano), Peter Bee (guitar) and Ricky Martyn (drums).
For Clairy Browne & the Bangin' Rackettes, contact:
Libby Coffey/MSO
LCoffey@msopr.com
For Vanguard Records, contact:
Lucy Sabini 310-829-9355 x137
lsabini@vanguardrecords.com
This entry was posted in CLAIRY BROWNE AND THE BANGIN' RACKETTES, Press Releases. Bookmark the permalink.


The intervention rebuilds the building phases of this house in a didactic way: the original core of the 40's of the 20th century and the small following additions. In the more ancient portion of the house the reconstructed axial ways distribute four rooms of the same size; the extension overlooking the garden is freed from the dividing inner walls to form living rooms above the two floors. The rediscovered succession of the works is displayed in the drawing of the fronts and their renewed decorations.


text:
Down the Shore
Barnegat Inlet jetty repairs begin shortly in Island Beach State Park
Justin Auciello
Fishermen cast from Island Beach State Park's Barnegat Inlet jetty in 2010. (Photo: Jennifer Harker/JMA Creations)
Construction crews will soon begin repairs on a significant portion of the north side of the Barnegat Inlet jetty that protects the southern tip of Island Beach State Park, a spokesman for the U.S. Army Corps of Engineers' Philadelphia District said today.
The $7.6 million project will result in repairs to approximately 740 feet of the jetty that was badly damaged by Superstorm Sandy, according to a February 2014 Army Corps release.
"Our contractor, Agate Construction Company, is currently mobilizing equipment to begin the repair work. All of the land-based equipment should be on site by March 15, while a barge will arrive in the inlet at a later date and be positioned alongside the jetty as they make repairs," Army Corps spokesman Steve Rochette said in an email.
The repair work involves rebuilding the cross-section of the jetty using existing capstone, supplemented with additional capstone, corestone, slushed concrete and stone-filled mattresses.
Beach buggy access will continue for permit holders, but the beach area near the jetty will be closed for the duration of the project, which is expected to continue to the fall, according to the release.
"While we recognize the jetty is a popular fishing destination, it is imperative that the jetty area be closed to all go park-goers for their own safety," said Mark Texel, New Jersey State Park Director. "Ultimately, we will have a stronger jetty that will best serve Barnegat Inlet and the south end of our park."
Brought to you by Down the Shore
About Justin Auciello
@auciello auciello@gmail.com